Product–market flexibility and capital structure

The Quarterly Review of Economics and Finance 54 (2014) 111–122

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Product–market flexibility and capital structure Sudipto Sarkar ∗ DSB 302, DeGroote School of Business, McMaster University, 1280 Main Street West, Hamilton, ON L8S 4M4, Canada

a r t i c l e

i n f o

Article history: Received 14 March 2012 Received in revised form 6 June 2013 Accepted 30 September 2013 Available online 10 October 2013 Keywords: Product–market flexibility Price sensitivity Capital structure Contingent-claim model

a b s t r a c t Many companies have the ability to adjust their product’s price and/or quantity in response to changes in the marketplace. We show that this product–market flexibility or market power, hitherto ignored in the contingent-claim modeling literature, can potentially have a significant effect on the corporate capital structure decision. When the firm is operating at full capacity, product–market flexibility is not important, hence market power has a negligible effect on optimal capital structure. However, when operating below capacity, product–market flexibility becomes important and market power has, in general, a positive effect on optimal debt level and optimal leverage ratio. This is consistent with available empirical evidence. Numerical results indicate that the effect of product–market flexibility on optimal debt level and optimal leverage ratio can potentially be large enough to be economically significant, hence it should not be ignored as a determinant of capital structure. © 2013 The Board of Trustees of the University of Illinois. Published by Elsevier B.V. All rights reserved.

1. Introduction Contingent-claim models are an important component of the theoretical capital structure literature in Corporate Finance. Starting with Brennan and Schwartz (1978), a large number of papers have examined the corporate leverage decision using contingentclaim models, e.g., Fischer, Heinkel, and Zechner (1989), Goldstein, Ju, and Lel (2001), Hackbarth and Mauer (2012), Leland (1994, 1998), Leland and Toft (1996), Mauer and Ott (2000), Mauer and Sarkar (2005), Sarkar and Zapatero (2003), Sundaresan and Wang (2007), Titman and Tsyplakov (2007), and many more. In all of these models, the source of uncertainty (state variable) is the exogenously specified earnings/cash flows or asset value of the firm. But this ignores an important managerial flexibility – the ability to adjust production as market (demand) conditions fluctuate. The only exception, to our knowledge, is Mauer and Triantis (1994), where the firm can shut down and restart operations when market demand changes sufficiently in either direction. However, since they do not allow the firm to adjust price or output level, their model’s view of product–market flexibility is somewhat limited. This paper contributes to the literature by incorporating product–market flexibility or market power in the capital structure decision. We extend the standard contingent-claim model by including the firm’s product–market power, i.e., price- and quantity-setting flexibility in the product market. This is an important issue because the available empirical evidence indicates that

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product–market power does, in general, have a significant effect on corporate capital structure decisions (Guney, Li, & Fairchild, 2011; MacKay & Phillips, 2005; Pandey, 2004; Rathinasamy, Krishnaswamy, & Mantripragada, 2000; Smith, Chen, & Anderson, 2008). While some papers have examined theoretically the interaction of product market and capital structure, they have been limited to strategic/game-theoretic models (Brander & Lewis, 1986; Lyandres, 2006; Maksimovic, 1988). Firms often have substantial flexibility in an uncertain product market, because of their ability to adjust output price and/or quantity as market conditions change. When demand for its product falls (rises), a company will reduce (increase) price and/or quantity, subject to its capacity limit.1 However, this product–market flexibility is ignored in the contingent-claim capital-structure models referred to above, which assume that earnings or asset value is an exogenous random process that is unaffected by the firm’s product–market decisions such as quantity and pricing.2 But since the firm generally has the ability to adjust output price and/or quantity, earnings should be endogenously determined. This endogeneity of earnings is incorporated in our model, which assumes a more primitive source of uncertainty – the exogenously specified random demand shock.

1 Thus, product–market flexibility will be particularly important when the firm’s capacity constraint is not binding, i.e., when it is operating below full capacity, since it is then free to change both output quantity and price. 2 Even when the product market is considered (e.g., Titman & Tsyplakov, 2007), it is assumed that the firm’s product market is described by an exogenous price process, and the firm always operates at full capacity, in which case it has no productmarket flexibility since it cannot adjust either output price or output quantity.

1062-9769/$ – see front matter © 2013 The Board of Trustees of the University of Illinois. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.qref.2013.09.002

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We start with a downward–sloping stochastic demand curve that the firm faces in the output market. Thus, the firm is operating in a differentiated-product market (monopolistic competition or monopoly), with a downward–sloping demand curve. The firm’s product–market flexibility will then be determined by demand elasticity (or price sensitivity), as discussed in Section 2. We therefore focus our attention on the effect of this parameter on optimal capital structure. We show that, even apart from strategic or game-theoretic considerations, product–market characteristics can have an important effect on corporate capital structure. When the firm is not operating at full capacity, the optimal debt level and optimal leverage ratio are both increasing functions of price sensitivity; the effect is insignificant when the firm is operating at full capacity, which is not surprising since, at full capacity, the firm does not have operating flexibility. Also, the optimal debt level is a U-shaped function, and optimal leverage ratio a decreasing function, of demand volatility. The rest of the paper follows the outline below. Section 2 describes the model and derives the firm’s earning stream and asset value under product–market flexibility. Section 3 examines the capital structure decision, and identifies the optimal debt level and optimal leverage ratio. Section 4 presents the results, and Section 5 summarizes and concludes.

2. The model We use a contingent-claim model to study the corporate capital structure decision, as in Leland (1994), Goldstein et al. (2001), Sarkar and Zapatero (2003). However, since we are interested in the effect of the company’s product’s demand characteristics, our state variable is the demand shock rather than earnings or asset value (as in the existing contingent-claim models). A firm produces q units of the output per unit time, which it sells at a price of $p per unit. The firm can set the price and quantity, subject to a couple of constraints. The first constraint is the capacity limit – we assume that the firm has a physical capacity limit of Q per unit time, thus it cannot produce output at a higher rate, i.e., qt ≤ Q, ∀t. The second constraint is the demand relationship – we assume that the firm faces a downward–sloping demand curve with constant elasticity (Dobbs, 2004): pt =

yt (qt /Q )



(1)

where the state variable yt represents a stochastic shock to demand. The parameter  indicates the price sensitivity to output quantity changes (0 ≤  ≤ 1). The demand curve (1) implies a constant price elasticity of demand of (1/). A higher value of  denotes greater market power for the firm, and  = 0 represents no market power (perfect competition) with an exogenous price process. The company incurs a constant variable cost of $v per unit produced. Its profits are taxed at a (symmetric) tax rate of  (that is, profits and losses are treated the same).3 Also, the company is

3 Thus, if taxable income is negative, there will be a positive cash flow from corporate tax (i.e., the company will receive a tax rebate from the government). This is not an exact depiction of the actual tax code, which is asymmetric; however, it is a standard assumption in the contingent-claim literature for reasons of tractability (Leland, 1994; Mauer & Ott, 2000; Sundaresan & Wang, 2007). Moreover, it is not a damaging assumption because, in real life, companies can carry forward tax losses and thereby benefit from their losses. In any case, we can take care of any remaining asymmetry in the tax code by reducing effective tax rate  appropriately in our computations (as discussed in Graham & Smith, 1999).

risk-neutral and discounts cash flows at the constant discount rate of r.4 The state variable y is the demand shock, which can be viewed as the strength of market demand, i.e., a higher y implies stronger demand. It introduces uncertainty in the model, and evolves according to a lognormal process (Aguerrevere, 2003; Dobbs, 2004): dy/y = dt + dz

(2)

where  and  are expected growth rate and volatility, respectively, of y; and z is a standard Weiner process. When  = 0 (perfect competition) the firm faces an exogenous price process pt = yt , and the output policy will be as follows: produce at the maximum rate (qt = Q) when yt > v, and nothing (qt = 0) when yt ≤ v. Thus, when  = 0, the firm has no discretion over either price or quantity, i.e., no product–market flexibility. Product–market flexibility is relevant only for  > 0. As  is increased, the firm has more market power, hence greater ability to benefit from the ability to adjust price and quantity. Thus, product–market flexibility is more valuable when  is higher. 2.1. The earnings stream with product–market flexibility We assume that the firm acts rationally, hence it adjusts price and quantity continuously in response to changes in demand (y) so as to maximize the profit stream (Aguerrevere, 2003; Sarkar, 2009).5 If it produces output at a rate of q units per unit time, and sells the output at a price of $p per unit, subject to Eq. (1), then the after-tax instantaneous profit is given by: (q) = (1 − )(p − v)q = (1 − )(yQ  q1− − vq) Setting d/dq = 0, we get the optimal quantity as a function of the 1/ state variable: qt = Q [yt (1 − )/v] , and the optimal output price: pt = v/(1 − ). The resulting profit stream is given by: (y) = (1 − )y1/

(3)

where  = Q [v/(1 − )]

1−1/

(4)

Differentiating (y) in Eq. (3) with respect to , it is easily shown that d(y)/d > 0. Therefore, when the capacity constraint is not binding (hence the firm has operating flexibility), the earnings level of the firm is an increasing function of the price sensitivity . This should not be surprising, since increased market power will result in higher profits. It is clear from the above that as the strength of demand (y) rises, the price remains unchanged but the output quantity increases accordingly. However, there is a capacity constraint q ≤ Q; hence the output quantity can increase only up to the limit Q. Suppose the 1/ ¯ then Q (y(1 ¯ − )/v) capacity limit is reached when y rises to y; = Q . Thus, the capacity-constraint boundary (i.e., the level of y at which full capacity is reached) is y¯ = v/(1 − ). For higher levels of y, the output cannot rise and is therefore unchanged at Q, but

4 Risk neutrality is a standard assumption in the literature (Aguerrevere, 2003; Dobbs, 2004). Risk aversion can be incorporated as suggested by Aguerrevere (2003, footnote 5), but it will add complexity without qualitatively changing the results. 5 Since the firm adjusts quantity continuously, the implicit assumption is that adjustment costs are zero. This is a common assumption in the Economics literature (Hagspiel, Huisman, & Kort, 2009; He & Pindyck, 1992; Sarkar, 2009), justified by the fact that reduction of output quantity is achieved simply by not using some of the installed capacity, and increase of output by using some more of the installed capacity. There are generally no significant costs attached to such increases/decreases in output. Adjustment costs would, however, be important if the production was to be temporarily shut down and restarted (Dixit, 1989; Mauer & Triantis, 1994).

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the price rises according to the demand curve (Eq. (1)). Thus, for ¯ the output level and output price are given by: q = Q, and y > y, p = y. The corresponding profit flow will be:

As shown in Appendix A, V(y) must satisfy the ordinary differential equation (ODE):

(y) = (1 − )(y − v)Q

where (y) is the profit flow derived above, given by (1 − )y1/ in region 1 and (1 − )(yQ − vQ) in region 2. Since (y) depends on the operating region, the project value V(y) will also depend on the operating region, hence it is derived separately for each region. ¯ Solving ODE (9) with (y) = (1 − )y1/ , we get: Region 1 (y ≤ y):

(5)

From Eq. (5), earnings level (y) is independent of . Therefore, when the capacity constraint is binding (hence no operating flexibility), price sensitivity has no effect on earnings. When capacity constraint is binding, increased market power does not translate into higher profit because the firm has no degrees of freedom (with capacity fixed at Q, the price just comes from the demand curve and the firm has no control over it), hence no product–market flexibility. To summarize, there are two operating regions: ¯ Region 1 (y ≤ y): p = v/(1 − ), q = Q [y(1 − )/v] (y) = (1 − )y1/ . ¯ p = y, q = Q and (y) = (1 − )(yQ − vQ). Region 2 (y > y):

1/

,

and

2.2. The dynamics of the earnings stream

dt +

 dz 

(6)

Thus the earnings process is also lognormal, but both the drift and volatility are functions of the demand sensitivity . Importantly (because earnings volatility affects optimal capital structure), we note that earnings volatility is a decreasing function of . A higher  denotes more market power; when the firm has greater market power (and recall that capacity constraint is not binding in this region), it is better able to adjust price and/or quantity and thus it can moderate the volatility effects of market demand. This reduces the volatility of the earnings stream when  is higher. 2.2.2. Region 2 Using Ito’s lemma to differentiate (y) in Eq. (5), we get after some simplification: d = [1 + (1 − )vQ/]dt + [1 + (1 − )vQ/]dz 

(1 − ) r − / − 0.5 2 (1/ − 1)/

(11)

A is a constant to be determined by the boundary conditions, and  1 is the positive root of the quadratic (12)

In Eq. (10), the first term is the value of operating forever without a capacity constraint (i.e., permanently in operating region 1), and the second term captures the possibility of the capacity constraint binding in the future (therefore, A < 0). ¯ Solving ODE (9) with (y) = (1 − )(yQ − vQ), Region 2 (y > y): we get:

 Qy

(r − )

−

vQ r



+ By2

(13)

In an infinite-horizon setting, the value of the firm’s assets (or the unlevered firm value) will be a function of the state variable y (and also the operating region). Thus, the value can be written as: in operating region1

V2 (y)

in operating region2

2

(14)

2

(15)

(0.5 − / 2 ) + 2r/ 2

 2

2 = 0.5 − / −

(0.5 − / 2 ) + 2r/ 2

In Eq. (13), the first term is the value of operating permanently in region 2, and the second term captures the possibility of the capacity constraint becoming non-binding in the future. To complete the valuation of the firm’s assets, the two constants A and B must be identified from the boundary conditions. There is ¯ Two one boundary here: the capacity-constraint boundary y = y. conditions must be satisfied at this boundary (Dixit & Pindyck, 1994; Leland, 1994): (a) the value-matching or continuity con¯ = V2 (y) ¯ and (b) the smooth-pasting or optimality dition: V1 (y) ¯ = V2 (y). ¯ The two boundary conditions give: condition: V1 (y) A=

¯ (1 − 1/2 )/(r − ) − vQ/r] − Z(y) ¯ 1/ (1 − 1/2 ) (1 − )[yQ ¯ 1 (1 − 1 /2 )(y) (16)

B=

¯ 1/ Z(1 − 1/1 ) − (1 − )[yQ ¯ (1 − 1/1 )/(r − ) − vQ/r] (y) ¯ 2 (1 − 2 /1 )(y) (17)

2.3. Unlevered firm valuation

V1 (y)

1 = 0.5 − / 2 +

(7)

In this region, the earnings process does not exactly inherit the demand variable’s lognormal characteristics. Note that the earnings volatility here is independent of the price sensitivity .

V (y) =

Z=



which simplifies to



(10)

where B is a constant to be determined by the boundary conditions, and  2 is the negative root of the quadratic Eq. (12). The two roots  1 (>0) and  2 (<0) are given by:

d = [(1 − )y1/ /][( + 0.5 2 (1/ − 1))dt + dz]



(9)

where

V2 (y) = (1 − )

2.2.1. Region 1 Differentiating (y) from Eq. (3) using Ito’s lemma, we get

d = 

V1 (y) = Zy1/ + Ay1

0.5 2 ( − 1) +  − r = 0

In Region 1 the firm is not forced to produce at a fixed output level, but has the ability to change output depending on the strength of demand. This represents product–market flexibility in our model. There is no product–market flexibility in Region 2.

 + 0.5 2 (1/ − 1)

0.5 2 y2 V  (y) + yV  (y) − rV (y) + (y) = 0

(8)

We have now completed the valuation of the firm’s assets. Unlike the case of earnings (Sections 2.1 and 2.2), the effect of  on the firm’s asset value V(y) and asset value volatility cannot be derived analytically, because of the complexity of the expressions involved. We therefore demonstrate the effect with numerical results in Section 4.1. Intuitively, we expect the following. In Region 1, since earnings are increasing in , value should also be an increasing function of . In Region 2, earnings are independent of , hence

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there should be no direct effect of  on value. However, there is a positive probability of moving from Region 2 to Region 1 (i.e., a positive probability that y will fall below y¯ in the future). Since asset value is also affected by future earnings expectations, it should also be an increasing function of  in Region 2. However, this is a secondorder effect, hence the effect of  on value should be much smaller in Region 2 than on Region 1. In a similar manner, the volatility of asset value should be a decreasing function of  in Region 1, since earnings volatility is a decreasing function of . In Region 2, earnings volatility is independent of ; however, since value depends on future earnings and there is a positive probability of moving to Region 1, the volatility of asset value in Region 2 should also be a decreasing function of , although the effect will be much smaller than in Region 1.

3.2. Equity valuation

3. The leverage decision

where B1 , B2 and B3 are constants to be determined from the boundary conditions. In Eq. (21), the first two terms on the right hand side represent equity value if the firm were to operate forever without bankruptcy or capacity constraint, and the other two terms represent the effects of the bankruptcy and capacity constraint boundaries, y = yb and y = y¯ respectively. In Eq. (22), the first term on the right hand side represents the equity value if the firm was to operate forever at full capacity, and the other term represents the effect of the boundary ¯ y = y. Thus, four unknowns (B1 , B2 , B3 and yb ) have to be determined from the boundary conditions. There are two boundaries: (i) bankruptcy boundary (y = yb ) and (ii) capacity-constraint boundary ¯ Once again, the value-matching and smooth-pasting condi(y = y). tions must be satisfied at each boundary. Keeping in mind that the payoff to shareholders at bankruptcy is zero, we get the following boundary conditions:

Now suppose the firm has debt in its capital structure. The level of debt financing is given by the coupon obligation of $c per unit time to bondholders. We assume it is long-term debt, i.e., the coupon payments will be made in perpetuity or until bankruptcy. The bankruptcy process is modeled as in the corporate finance literature (Leland, 1994; Mauer & Ott, 2000). The company will file for bankruptcy when the state variable y falls to a sufficiently low level (say, y = yb ). At bankruptcy, the company will incur a fractional bankruptcy cost of ˛ (1 ≥ ˛ ≥ 0), its assets will be acquired by the debt holders, and the equity holders will be left with nothing. 3.1. Debt valuation The firm’s debt is valued in a similar manner to project valuation, by solving the ODE (9) with (y) replaced by c (since the cash flow stream to debt holders consists of the coupon payment) and the appropriate boundary conditions. Since the cash flow to debt holders will be the same in both operating regions, the debt valuation formula, D(y), will be independent of the operating region. Solving ODE (9) with D(y) instead of V(y), and using (y) = c, we get: D(y) = c/r + D1 y2

(18)

where D1 is a constant to be determined from the boundary condition. The term c/r represents the value of the debt if there was no bankruptcy risk, and the other term represents the bankruptcy risk (hence D1 < 0). As mentioned above, at the bankruptcy boundary (y = yb ) the payoff to bondholders is (1 − ˛) times the unlevered firm value. The formula for unlevered firm value at bankruptcy (i.e., V1 (yb ) or V2 (yb )) will depend on whether the firm is operating at full or partial capacity when it declares bankruptcy. It does not make sense to declare bankruptcy when demand is so high that the firm is operating at full capacity (unless the debt level is extremely high, but that would not be optimal, and this paper is concerned with optimal debt levels), hence for practical purposes the bankruptcy trigger will not be in Region 2. We therefore assume the bankruptcy trig¯ b .6 The value of the firm’s assets at ger will be in Region 1, i.e., y≥y bankruptcy will therefore be V1 (yb ), giving the boundary condition: D(yb ) = (1 − ˛)V1 (yb ), from which we get the constant D1 = [(1 − ˛)Z(yb )1/ + (1 − ˛)A(yb )1 − c/r](yb )−2

(19)

The bankruptcy trigger yb is determined below.

6 The other case (i.e., extremely large debt level and bankruptcy in region 2) can be handled in the same manner, but is of no practical importance, since the optimal ¯ b ). debt level (Section 4) is always such that the bankruptcy trigger is in Region 1 (y≥y

If the equity value is given by E(y), we can write:

 E(y) =

E1 (y)

in operating region1

E2 (y)

in operating region2

(20)

since equity value will be state-contingent. It can be shown, as in the earlier cases, that E1 (y) = Zy1/ − (1 − )c/r + B1 y1 + B2 y2

(21)

E2 (y) = (1 − )[Qy/(r − ) − (vQ + c)/r] + B3 y2

(22)

• At y = yb : Value-matching: E1 (yb ) = 0, Smooth-pasting: E  (yb ) = 0. 1 • At y = y: ¯ Value-matching: E1 (y) ¯ = E2 (y), ¯ Smooth-pasting: E1 (y) ¯ = ¯ E2 (y). From the boundary conditions, we get the three constants: B1 =

¯ − 1/2 )/(r − ) − vQ/r] − Z(1 − 1/2 )(y) ¯ 1/ ] (1 − )[Q y(1 ¯ 1 (1 − 1 /2 )(y) (23)

B2 = −(yb )1/−2 Z/(2 ) − B1 (1 /2 )(yb )1 −2

(24)

¯ 1 −2 + Z(y) ¯ 1/−2 −(1 − )[Q y/(r ¯ ¯ −2 B3 = B2 + B1 (y) − )−vQ/r](y) (25) Note that B2 and B3 are functions of the optimal bankruptcy trigger yb , which is determined as the solution of an implicit equation: (1 − 1/2 )Z(yb )1/ − (1 − )c/r + B1 (yb )1 (1 − 1 /2 ) = 0

(26)

Eq. (26) has no analytical solution and has to be solved numerically for yb . We have now completed the valuation of the firm’s debt and equity, D(y) and E(y) respectively. The total value of the firm is the sum of equity and debt: F(y) = D(y) + E(y), and is given by:

 F(y) =

1

Zy1/ + c/r + B1 (y)

2

+ (B2 + D1 )(y)

in Region1 2

(1 − )[yQ/(r − ) − vQ/r] + c/r + (B3 + D1 )(y)

in Region2

(27)

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3.3. Optimal capital structure

4.2. Effect of price sensitivity  on asset value V(y)

When issuing debt, the firm must decide on the debt level optimally, i.e., so as to maximize the total value of the firm (equity plus debt); see Leland (1994). Thus, the optimal debt level is given by:

The effect of price sensitivity  on earnings (y) has been discussed in Section 2.1: in Region 1 (capacity constraint not binding) earnings level is an increasing function of , in Region 2 (capacity constraint binding) earnings level is independent of . The effect of  on asset value or unlevered firm value, V(y), is a little more complicated because the asset value in any operating region is affected (via the boundary conditions) by the possibility of moving to the other operating region. Fig. 1 shows V(y) as a function of . The results are computed with the base-case parameter values specified in the previous subsection (and y = 2), as well as for a range of parameter values. Since v = 1, the capacity-constraint boundary is y¯ = 1/(1 − ). The firm is at the capacity-constraint boundary when  = 0.5 (since y = 2), which is shown in the figures by the vertical broken line. To the left ¯ hence the firm is of this boundary (i.e., for  < 0.5) we have y > y, operating in Region 2. To the right of the vertical line ( > 0.5), we ¯ hence the firm is operating in Region 1. have y < y, In all cases, asset value is an increasing function of . In Region 1 (where the firm has operating flexibility)  has a significant effect on value, while in Region 2 (where the firm has no operating flexibility) the effect is generally insignificant. Both relationships are consistent with economic intuition, as discussed in Section 2.3. Although the effect of  on firm’s asset value could not be shown analytically, numerical results indicate the same relationship for all parameter values examined, hence we are confident of the generality of the result.

c∗ = Argmax F(y, c) c

where we write the total firm value as F(y,c) to make it clear that the coupon (debt) level is a choice variable. Setting dF(y,c)/dc = 0, gives us the first order condition for optimal debt level c* :  + (y)2 r

 dD

1

dc

+

dB2 dc



=0

(28)

where [Z(2 − 1/)(yb )1/ − B1 1 (1 − 2 )(yb )1 ] dB2 = dc 2 (yb )2 +1 

dyb dc

[c2 + (1 − ˛)rA(1 − 2 )(yb ) 1 + r(1 − ˛)Z(1/ − 2 )(yb ) dD1 =  +1 dc r(yb ) 2

,

1/ dyb ] dc

− yb

,

(1 − )yb dyb . = dc rZ(1 − 1/2 )(1/ − 1 )(yb )1/ + (1 − )c1 Eq. (28) also has to be solved numerically, as it has no analytical solution.7 To jointly identify the optimal debt level (c* ) and bankruptcy trigger (yb ), we solve Eqs. (26) and (28) simultaneously. The resulting optimal leverage ratio is then given by D(y,c* )/F(y,c* ). 4. Results The model is too complex to allow for analytical solutions, hence the results are illustrated with numerical solutions. For numerical results, the values of the various input parameters must be specified. We start with a “base case” set of parameter values that are reasonable and based on well-known papers in the existing literature.8 However, all computations are repeated numerous times with various parameter values, to ensure that the results are robust and not dependent on the parameter values chosen. 4.1. Base-case parameter values For the “base case,” we set the plant capacity at Q = 1 and the variable operating cost per unit at v = 1; these are without any loss of generality, since they are just scaling variables. For the other variables, we base the values on Leland (1994), which is a seminal paper in the contingent-claim capital-structure literature. Thus, we set the tax rate at  = 15% (see footnote 27 of Leland, 1994), bankruptcy cost ˛ = 50%, interest rate r = 6%, and volatility  = 20%. The demand growth rate is set at  = 1%. We use these parameter values to solve for the optimal capital structure with different levels of price sensitivity (), the parameter associated with product–market flexibility.9

7 Note that the first order condition (28) is the same for both operating regions ¯ since dB1 /dc = 0 and dB2 /dc = dB3 /dc. (y ≤ y¯ and y > y) 8 This is true for most variables, but the price sensitivity () has not been examined in the existing literature, hence there is no precedent for us to rely on. For this variable, we display the results for the reasonable range of values of the variable, e.g.,  = 0.05–0.95. 9 In Appendix B, we present numerical results with parameter values from some other well-known papers; the results are found to be very similar.

Result 1. In Region 1 (capacity constraint not binding, hence firm has operating flexibility), earnings level and asset value are both increasing functions of price sensitivity; in Region 2 (capacity constraint binding, hence no operating flexibility) earnings level is independent of, and asset value is slightly increasing in, price sensitivity. 4.3. Optimal capital structure To illustrate a specific case, let us identify the optimal capital structure with the base-case parameter values, price sensitivity  = 0.5 (implying a price elasticity of 2), and y = 2. The capacity¯ i.e., the firm is constraint boundary is y¯ = v/(1 − ) = 2; thus y = y, at the boundary of the two operating regions. With these parameter values, the solution to Eqs. (26) and (28) gives the following output: Optimal debt level c ∗ = 0.6599, optimal bankruptcy trigger yb = 0.7243, and optimal leverage ratio = 40.41%.

4.4. Effect of price sensitivity  Fig. 2 illustrates the effect of  on the optimal coupon level (c* ) and the optimal leverage ratio with the base case parameter values, for three levels of demand volatility,  = 10%, 20% and 30%. The vertical broken line ( = 0.5) shows the boundary between operating ¯ To its left ( < 0.5), the firm is operating in regions 1 and 2, i.e., y = y. ¯ to its right ( > 0.5), Region 2 (no operating flexibility), since y > y; the firm is operating in Region 1 (with operating flexibility), since ¯ y < y. There are three points worth noting. One, the sensitivity of the leverage decision to  is greater when demand volatility  is smaller. This is not surprising, since  should have a (relatively) bigger impact on asset volatility when  is smaller. Two, the effect

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30

30

Asset value

Asset value

27 24

25

20

21

15

18 0

0.2

0.4

0.6

0.8

1

0.0

0.2

Price sensitivity

0.4

0.6

0.8

1.0

Price sensitivity

Sigma=15%

Sigma=20%

Mu=0.5%

Mu=1%

Sigma=25%

Boundary

Mu=1.5%

Boundary

(a)

(b) 30

40

Asset value

Asset value

27

30

20

24

21 18 0.0

10 0

0.2

0.4

0.6

0.8

1

0.2

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1.0

Price sensitivity

Price sensitivity r=5%

r=6%

r=7%

Boundary

(c)

Tax=10%

Tax=15%

Tax=20%

Boundary

(d)

Fig. 1. Shows firm’s asset value (or unlevered firm value) V(y) as a function of price sensitivity . Part (a) shows the relationship for different levels of volatility or sigma (), part (b) for different growth rates (), part (c) for different interest rates (r), and part (d) for different tax rates (). The base-case parameter values are used:  = 1%,  = 20%, r = 6%,  = 15%, ˛ = 50%, v = 1 and Q = 1. The broken vertical line shows the boundary between operating regions 1 and 2.

of  depends on whether the firm is operating at full or partial capacity when making the capital structure decision. Three, both optimal debt level c* and optimal leverage ratio are increasing functions of  in operating region 1, but insensitive to  in operating region 2. Thus, the traditional contingent-clam models without product–market flexibility can approximate reasonably well companies operating at full capacity, but not companies operating below capacity. Operating Region 2 ( < 0.5): For low volatility ( = 10%), both c* and optimal leverage ratio are initially increasing, then decreasing, and finally increasing, in ; for moderate volatility ( = 20%), both c* and optimal leverage ratio are virtually independent of ; and for higher volatility ( = 30%), both c* and optimal leverage ratio are slightly increasing in . In all cases, the effect of  on debt level and capital structure is negligible. Operating Region 1 ( > 0.5): Both c* and optimal leverage ratio are in all cases increasing functions of , and the effect is not insignificant.

These relationships can be explained as follows. A higher price sensitivity  results in a lower asset volatility (the “volatility” effect), from Section 2.3. The lower asset volatility has two effects: (i) from standard option theory, lower volatility brings closer the bankruptcy boundary (i.e., raises the bankruptcy trigger yb ), which increases bankruptcy risk and makes debt less desirable; and (ii) lower volatility also means that extreme realizations (both high and low) are less likely, which reduces the likelihood of reaching the bankruptcy trigger; this reduces bankruptcy risk and makes debt more desirable. The net effect of  on the optimal leverage ratio will depend on which of the above two effects dominates. In general, the second effect seems to dominate, hence optimal leverage ratio is generally an increasing function of . Recall from Section 2.3 that  has a negligible effect on asset volatility in Region 2 but a significant effect in Region 1. Therefore,  should also have a significant effect on optimal leverage ratio in Region 1, but a minor effect in Region 2. This is indeed what we find, as illustrated in Fig. 2 – optimal leverage ratio is an

S. Sarkar / The Quarterly Review of Economics and Finance 54 (2014) 111–122

Therefore,  has an additional positive effect on c* that is negligible in Region 2 but significant in Region 1. This is indeed what we find, as Fig. 2 illustrates – c* behaves very similar to optimal leverage ratio in Region 2 but rises much faster in Region 1 because of the “value” effect. We state below the main result of our paper, summarizing the effect of market power/price sensitivity :

Optimal coupon level

1.2

1

0.8

Result 2. The effect of price sensitivity on capital structure will depend on whether the capital structure decision is made when operating at full or partial capacity. When the firm is operating at full capacity, price sensitivity does not have a substantial effect on the firm’s capital structure; when operating below full capacity, optimal debt level c* and optimal leverage ratio are both increasing functions of price sensitivity.

0.6

0.4 0

0.2

0.4

0.6

0.8

1

Price sensitivity Sigma=10%

Sigma=20%

Sigma=30%

Boundary

(a) 0.7

Optimal leverage ratio

117

0.6 0.5 0.4 0.3 0.2 0

0.2

0.4

0.6

0.8

1

Price sensitivity Sigma=10%

Sigma=20%

Sigma=30%

Boundary

(b) Fig. 2. Shows optimal coupon level (c* ) and optimal leverage ratio as functions of price sensitivity , for three levels of volatility,  = 10%, 20% and 30%. The base-case parameter values are used:  = 1%, r = 6%,  = 15%, ˛ = 50%, v = 1 and Q = 1. The broken vertical line shows the boundary between operating regions 1 and 2.

increasing function of  in Region 1, but not very sensitive to  in Region 2. However, the exception is the case of low demand volatility ( = 10%), in which case there is a region (intermediate ) over which the optimal leverage ratio is actually a decreasing function of . Thus, for intermediate values of , the first effect seems to dominate. Therefore, for small demand volatility, optimal leverage ratio is initially an increasing function, then a decreasing function, and finally an increasing function again, of . For the optimal coupon level c* , there is an additional effect: a higher  results in higher asset value (the “value” effect), as discussed in Sections 2.3 and 4.2.10 As illustrated in Fig. 1, the value effect is negligible in Region 2 but significant in Region 1.

10 The “value” effect is not relevant for optimal leverage ratio because, when asset value increases, more debt is used but equity value also rises; thus, in spite of the higher debt level, the optimal leverage ratio is independent of asset value changes.

4.4.1. Magnitude of the effect For modeling purposes, a very important question is whether the effect of  on optimal leverage ratio is large enough to be economically meaningful. If not, there is not much to be gained by taking into account product market flexibility or price sensitivity. To answer this question, we can look at how the optimal leverage ratio changes when  is increased. With the base-case parameter values, when  is increased from 0.05 to 0.95, the optimal leverage ratio rises from 39.94% to 45.64%; that is, an increase in leverage ratio of 5.7%. This is certainly not an insignificant increase. The effect of  on capital structure is seen more clearly if we split it into the two operating regions. Within Region 2 (when  is increased from 0.05 to 0.5), optimal leverage ratio rises by only 0.47% (from 39.94% to 40.41%), but within Region 1 (when  is increased from 0.5 to 0.95), optimal leverage ratio rises by 5.23% (from 40.41% to 45.64%). Clearly, the effect of  is concentrated in operating region 1. For a firm operating in region 2, the effect of  is negligible. Table 1 summarizes the results with different parameter values. In this table, Lev1 is the spread in optimal leverage between  = 0.05 and 0.95, i.e., (optimal leverage ratio with  = 0.95) minus (optimal leverage ratio with  = 0.05). This measure shows how much the optimal leverage ratio can potentially change because of differences in price sensitivity. Lev2 and Lev3 show how this is split between operating region 2 and operating region 1 respectively, i.e., Lev2 is given by (optimal leverage ratio with  = 0.5) minus (optimal leverage ratio with  = 0.05), and Lev3 is (optimal leverage ratio with  = 0.95) minus (optimal leverage ratio with  = 0.5). Thus, Lev2 illustrates the effect of  in operating region 2 (at full capacity) and Lev3 illustrates the effect of  in operating region 1 (below full capacity), while Lev1 shows the overall effect of  on optimal leverage ratio. The spread in optimal leverage ratio varies depending on the parameter values, but is generally economically significant. While the range is 3.3–8.4% for the parameter values shown in the table, in most cases it is close to 5%. Thus, incorporating market power in the model can potentially result in a significantly higher optimal leverage ratio. When the firm is operating below full capacity, market power can make a material difference to the capital structure decision. Clearly, the effect of  on capital structure can be economically significant, and can result in substantially higher levels of debt usage than traditional models. In operating region 2, the firm is operating at full capacity and has no operating flexibility, hence product–market effects are not important and can be ignored. In this region, the effect of  on optimal leverage ratio is insignificant, hence the existing contingent-claim models without product–market flexibility

It is, in fact, a general result in contingent-claim models that optimal leverage ratio is independent of asset value (see, for instance, Leland, 1994).

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Table 1 Effect of price sensitivity () on optimal leverage ratio. Lev1

Lev2

Lev3

 15% 20% 25%

5.00% 5.70% 5.31%

−1.59% 0.47% 1.29%

6.59% 5.23% 4.02%

r 5% 6% 7%

5.55% 5.70% 5.63%

0.94% 0.47% −0.12%

4.61% 5.23% 5.75%

 0% 1% 2%

5.79% 5.70% 5.03%

−0.09% 0.47% 0.65%

5.88% 5.23% 4.38%

 10% 15% 20%

8.44% 5.70% 3.29%

2.08% 0.47% −0.99%

6.36% 5.23% 4.28%

˛ 40% 50% 60%

4.04% 5.70% 6.99%

−0.51% 0.47% 1.22%

4.55% 5.23% 5.77%

Base case parameter values: r = 6%,  = 1%,  = 20%,  = 15%, ˛ = 50%, Q = 1, and v = 1. Note: Lev1 : optimal leverage ratio with  = 0.95 minus optimal leverage ratio with  = 0.05. Lev2 : optimal leverage ratio with  = 0.5 minus optimal leverage ratio with  = 0.05. Lev3 : optimal leverage ratio with  = 0.95 minus optimal leverage ratio with  = 0.5.

are a reasonable approximation. To summarize, the effect of product–market flexibility on capital structure is (i) negligible for firms operating at full capacity, but (ii) potentially significant for firms operating below full capacity. Thus, there is an additional determinant of capital structure that must be taken into account in these models. However, we certainly do not claim that market power will always be a significant factor in capital structure decisions. The effect is very much a function of the circumstances facing the firm when it makes the capital structure decision, e.g., at full capacity market power will be insignificant in the capital structure decision. The best we can say from the model’s outputs is that in some situations, market power can have a significant effect on optimal debt level and optimal leverage ratio. This is, in our opinion, enough reason to consider including this factor in models of capital structure. From Table 1, we also note that the effect of  on optimal leverage ratio is particularly strong for certain parameter values (small ,  and large ˛). Therefore, it is more important to include this factor in the model for low-growth firms (e.g., old companies), companies with low effective tax rate (e.g., with large tax loss carryforwards), or companies with high bankruptcy cost (e.g., with a large percentage of intangible assets). As briefly referred to in Section 4.1, we follow Leland (1994) in setting the base-case tax rate at  = 15%. This is based on Miller’s (1977) argument that in a “trade-off” model where optimal capital structure is determined by trading off the tax advantage of debt versus the bankruptcy cost resulting from debt, the company’s effective tax rate is the net tax advantage of debt. Miller (1977) shows that the net tax advantage is given by the expression {1 − (1 −  c )(1 −  e )/(1 −  d )}, where  c ,  e , and  d are the corporate tax rate, tax rate on equity income, and tax rate on debt income, respectively. Using the values in Leland (1994, footnote 27), we get {1 − (1 − 0.35)(1 − 0.20)/(1 − 0.40)} = 0.133. For our base case, therefore, we use a slightly higher effective tax rate of  = 15%.

Also, since these capital structure models are infinite-horizon or perpetual models, a forward-looking long-term estimate should always be used. As mentioned by Leland (1994) and discussed in more depth by Graham and Smith (1999), the actual longterm forward-looking effective tax rate is likely to be even smaller because of real-life factors such as tax loss carryforwards, possibility of making losses in the future, etc. Finally, we have assumed, in common with the contingent-claim literature, that the tax rate is symmetric (i.e., profits are taxed and losses are subsidized). However, the tax rate in real life is asymmetric: while profits are taxed, losses are not subsidized. This introduces non-linearities in the effective tax schedule, which further reduces the effective tax rate (Graham & Smith, 1999). Because of these various reasons, it is possible (even likely) that the actual effective tax rate for many firms is lower than the base-case value we use, hence our base-case results could have understated the importance of market power in the capital structure decision. 4.4.2. Empirical implications The first empirical implication of Result 2 is that the effect of  depends on whether the company is operating at full capacity or below capacity when it makes the capital structure decision. This is a new empirical implication, since this aspect has not been examined in the literature; while there have been a number of empirical studies on the relationship between market power and leverage ratio, the difference between firms operating at full and partial capacity has not been examined. Testing this implication might be a challenge because the necessary information is not easily available from conventional data sources. The next empirical implication is that the relationship between market power and leverage ratio might be monotonic or nonmonotonic (U-shaped). In the context of empirical testing, a non-monotonic relationship would introduce non-linearity in the regression specifications. That is, if a regression is run with leverage ratio as the dependent variable, and market power (appropriately measured) as an explanatory variable, a positive and significant coefficient on market power would indicate that leverage is an increasing function of market power. However, with both market power and market power squared as additional explanatory variables, if the coefficients are negative and positive, respectively, it would indicate a U-shaped relationship. The most important empirical implication from Result 2 is that leverage ratio is generally an increasing function of market power. This is consistent with a number of empirical studies. Guney et al. (2011) provide evidence that leverage is positively related to market power, and in most cases the relationship is statistically significant. Rathinasamy et al. (2000) report that, for most of their samples, firms with higher monopoly power use more debt in their capital structure. Lovisuth (2008) reports a positive relationship between market power and leverage for most of the East Asian economies examined in the study, while Smith et al. (2008) report the same relationship for firms in New Zealand. Istaitieh and Rodriguez (2002) find that firms in highly concentrated industries tend to have higher debt levels. Since greater concentration generally implies less competition or more market power, their evidence is consistent with our prediction. Finally, MacKay and Phillips (2005) provide empirical evidence that firms in concentrated industries have higher leverage ratios than firms in competitive industries, which, in the context of our model, means that leverage ratio is an increasing function of . A further empirical implication is that, for low demand volatility, leverage ratio is a U-shaped function of market power. A few papers have found a U-shaped relationship, although these studies did not explicitly require their samples to be composed of only

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1.0

Optimal coupon level

low-volatility firms; hence they are not necessarily direct tests of this implication. Guney et al. (2011) report a U-shaped relationship between leverage ratio and market power in certain cases, with a negative coefficient on market power and a positive coefficient on market power squared. Pandey (2004) finds that for lower and higher market power debt usage is high, and for intermediate market power debt usage is low; this is also consistent with a U-shaped relationship. Finally, the model also implies that in certain cases, market power might have a negative effect on leverage ratio (the downward–sloping portion of the relationship). This is also consistent with some empirical findings. For instance, Rathinasamy et al. (2000) carried out industry-wise regressions, and reported that for some industries, debt usage is negatively related to monopoly power. Lovisuth (2003) also found a negative relationship between market power and debt usage in the U.K. retailing industry.

11 Earnings volatility and demand volatility may be related, but they are not the same, as shown in Section 2.2.

0.9

0.8

0.7

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0.5 0.0

0.2

0.4

0.6

0.8

Sigma Pr. Sens. = 0.05

4.5. Effect of demand volatility 

Pr. Sens. = 0.5

Pr. Sens. = 0.75

(a) 0.6

Optimal leverage ratio

The effect of earnings volatility or asset-value volatility on capital structure has been examined in existing papers, e.g., Leland (1994), Fischer et al. (1989), and Sarkar and Zapatero (2003). This section, however, looks at the effect of demand volatility on capital structure.11 Demand volatility  will affect bankruptcy risk in two ways, similar to the discussion in Section 4.4. First, from standard option theory, a higher  will move the bankruptcy trigger further (lower), thereby reducing bankruptcy risk. On the other hand, a higher  means that extreme values (both high and low) are more likely to be reached, hence the bankruptcy trigger is more likely to be reached; this increases bankruptcy risk. In addition to the “bankruptcy risk” affect,  affects debt choice via the “flexibility” channel. A higher  makes the embedded option (associated with product–market flexibility) more valuable, again from standard option theory. This increases value, hence it also affects the choice of debt level. Note, however, that there is no operating flexibility in Region 2; hence this effect will apply only to Region 1. The overall effect will depend on which of the above effects dominates. As discussed in Section 4.4, for small  and , the first effect seems to dominate; thus, when  is small, bankruptcy risk will initially be a decreasing function of , hence it will initially be optimal to take on more debt as  is increased. Thus, for small , c* should initially be an increasing function of . As  is increased further, the second effect starts to dominate, hence bankruptcy risk starts rising, making debt less attractive. Thus, c* becomes a decreasing function of . Eventually, the third effect mentioned above becomes more important, hence a higher  results in greater flexibility value and makes debt more attractive; thus the optimal debt level c* starts rising again as  is increased. To sum up, for small  the optimal debt level c* should be initially an increasing function, then a decreasing function, and finally an increasing function again, of . For other values of , since the first effect is not important, only the last two segments should be observed, hence c* should just be a U-shaped function of . For optimal leverage ratio, the third effect above does not apply, as discussed in Section 4.4 and footnote 10. Thus, for small , optimal leverage ratio will be initially increasing and subsequently decreasing in , to give an inverted-U shaped relationship; for all other values of , optimal leverage ratio will be a decreasing function of .

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0.5

0.3

0.2 0.0

0.2

0.4

0.6

0.8

Sigma Pr. Sens. = 0.05

Pr. Sens. = 0.5

Pr. Sens. = 0.75

(b) Fig. 3. Shows optimal coupon level (c* ) and optimal leverage ratio as functions of volatility or sigma (), for three levels of price sensitivity,  = 0.05, 0.5 and 0.75. The base-case parameter values are used:  = 1%, r = 6%,  = 15%, ˛ = 50%, v = 1 and Q = 1.

Fig. 3 illustrates how demand volatility  affects optimal coupon c* and optimal leverage ratio. It can be seen that the results are just as expected form the discussion above. The effect of  is summarized below. Result 3. The optimal coupon level is a U-shaped function of demand volatility, and the optimal leverage ratio is a decreasing function of demand volatility, except when  is very small, in which case the optimal coupon is first increasing, then decreasing, and finally increasing, in , while optimal leverage ratio is initially increasing and then decreasing in . Fig. 4 shows the relationship between c* and  for different parameter values. The same base-case parameter values are used as in Section 4.1, along with  = 0.5 (since the shape of the relationship does not vary with , with the exception of very small , as seen above). It can be noted that the relationship is more or less the same for all the parameter values examined. The only thing that varies is the exact point at which c* starts rising as  is increased. When demand growth rate  is higher, the firm will more likely operate at full capacity, hence product–market flexibility becomes

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1.00 Optimal coupon level

Optimal coupon level

1.4 1.2 1

0.8 0.6 0.4 0

0.2

0.4

0.6

0.90 0.80 0.70 0.60 0.50 0.00

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0.20

Sigm a Mu=0

0.60

0.80

Sigm a

Mu=1%

r=5%

Mu=2%

(a)

r=6%

r=7%

(b)

1.4

1.2

1.2

Optimal coupon level

Optimal coupon level

0.40

1 0.8 0.6 0.4 0.2 0

1

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0

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Sigm a Tax=10%

Tax=15%

0.4

0.6

0.8

Sigm a

Tax=20%

(c)

Alfa=40%

Alfa=50%

Alfa=60%

(d)

Fig. 4. Shows optimal coupon level (c* ) as a function of volatility or sigma () for different parameter values. Part (a) shows the relationship for different growth rates (), part (b) for different interest rates (r), part (c) for different tax rates (), and part (d) for different levels of bankruptcy cost (˛). The base-case parameter values are used:  = 1%, r = 6%,  = 15%, ˛ = 50%, v = 1 and Q = 1.

less important. Since the flexibility is less important, c* starts rising later (i.e., when  is higher). This is exactly what we observe in Fig. 4(a). The interest rate r has virtually no effect on the shape of the c* vs.  curve. When the tax rate  is higher, the after-tax cash flows from operations will be smaller, leading to a drop in the value of operating assets. Therefore, the value of product–market flexibility will rise relative to the value of operating assets, and flexibility will become more important. As a result, c* will start rising earlier (i.e., when  is smaller) when  is increased, and will start rising later when  is lower. This is also consistent with Fig. 4(c). In fact, for  = 10%, we note that c* is a decreasing function of  for the entire range examined. Finally, when bankruptcy cost ˛ is higher, the loss in firm value at bankruptcy is larger. Bankruptcy risk, therefore, becomes more important. As a result, c* will start rising later (when  is higher). Once again, this is exactly what we observe in Fig. 4(d).

5. Summary and conclusions Contingent-claim models of corporate capital structure assume exogenous earnings or asset values. However, earnings in real life are endogenous because they are determined by the firm’s activities in the product market; for instance, when demand for the product is stronger (weaker), a firm will often increase (reduce) its output and/or price. Its ability to make these adjustments (i.e., its market power) could then be a factor in determining the optimal capital structure. Empirical studies confirm that this is indeed the case. We therefore extend the traditional contingent-claim literature by incorporating the firm’s market power in the capital structure decision. This is the main contribution of our paper. We show that market power generally increases the optimal leverage ratio, consistent with the available empirical evidence. We also show that, in various situations, this relationship might be negligible, negative, or non-monotonic (U-shaped).

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If the firm makes the leverage decision when operating at full capacity, market power has no significant effect on the optimal leverage ratio, since the capacity constraint limits its ability to make adjustments. When operating below full capacity, however, we show that taking into account market power could increase the optimal leverage ratio by an economically significant amount. It is therefore not advisable to ignore this factor in models of corporate capital structure. A critical question in this context is: do firms make capital structure decisions before reaching full capacity? If the answer is no, then product–market flexibility can safely be ignored in the analysis of optimal capital structure. However, we contend that firms often do decide on their capital structure before reaching full capacity. This is because it usually takes a long time to reach full capacity, and firms do not generally defer capital structure decisions for such long periods. Moreover, it is often optimal to invest in a project before it can run at full capacity (Sarkar, 2009); in such cases, the capital structure decision will be made when operating at partial capacity. We therefore argue that the effect of product–market flexibility should be taken into account when identifying the optimal capital structure. We have essentially extended models such as Leland (1994), which examines a one-time capital structure decision. In practice, firms might change their capital structures (although not frequently, because changing the capital structure is a costly process, Faulkender, Flannery, Hankins, & Smith, 2007). This scenario has been examined by Fischer et al. (1989) and Goldstein et al. (2001) without product–market flexibility. Therefore, a possible extension of our model is to examine corporate capital structure decisions with product–market flexibility when recapitalization is possible.

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r. Equating the two for the period dt, we get: E(dV) + (y)dt = rVdt, which gives the ordinary differential Eq. (9) of the paper. Region 1 valuation: Here we solve ODE (9) with (y) = (1 − )y1/ (from Eq. (3)). The general solution to this ODE is y . Substituting this in the ODE, we get the quadratic equation: 0.5 2 ( − 1) +  − r = 0, which has two solutions  1 (>0) and  2 (<0). Thus, the complete general solution is: V1 (y) = Ay 1 + A2 y 2 , where A1 and A2 are constants to be determined by the boundary conditions. The particular solution, verified by direct substitution, is Zy1/ , where Z is given by Eq. (11). Thus, the complete solution is: V1 (y) = Ay1 + A2 y2 + Zy1/

(A3)

when y = 0, it will remain at zero forever (since y = 0 is an absorbing boundary), hence the asset value will also fall to zero. This gives the condition: Lim V1 (y) = 0, which implies A2 = 0 (since ␥2 < 0). Thus, y→0

we get Eq. (10) of the paper. Region 2 valuation: Here we solve ODE (9) with (y) = (1 − )(yQ − vQ) (from Eq. (5)). The general solution is the same as above, but the particular solution is (1 − )[Qy/(r − ) − vQ/r]. Thus the complete solution is:





V2 (y) = (1 − ) Qy/(r − ) − vQ/r + By2 + B1 y1

The term (1 − )[Qy/(r − ) − vQ/r] in Eq. (A4) is the value of the firm’s assets if it always operates at full capacity, with no possibility of operating flexibility any time in the future. When y reaches very high levels (y → ∞), it is very unlikely that the firm will ever go back to operating at partial capacity, hence Lim V2 (y) = (1 − )[Qy/(r − ) − vQ/r] y→0

Acknowledgement Financial support from a McMaster Innovation Grant is gratefully acknowledged. I would also like to thank two anonymous referees for their helpful comments and suggestions, and the Editor H.S. Esfahani for his recommendations during the review process. The usual disclaimer applies. Appendix A. We use the valuation approach of Dixit and Pindyck (1994), BarIlan and Strange (1999). V(y) is the value of the firm’s assets (or the unlevered firm value). These assets generate a cash flow of (y) in perpetuity. The expected increase in value after an infinitesimally small period of time (dt) is given by E(dV). From Ito’s lemma, we have: dV = Vy dy + 0.5Vyy (dy)

2

(A5)

Eq. (A5) implies that B1 = 0 in Eq. (A4), since  1 > 0. Thus, we get Eq. (13) of the paper. Derivation of Eq. (28): The total firm value in Region 1 is given by Eq. (27): F1 (y) = Zy1/ + c/r + B1 (y)1 + (B2 + D1 )(y)2 Differentiating this with respect to c, and noting that B1 is independent of c, we get Eq. (28) of the paper. For Region 2, total firm value is (also from Eq. (27)): F2 (y) = (1 − )[yQ/(r − ) − vQ/r] + c/r + (B3 + D1 )(y)2 Differentiating this with respect to c also gives Eq. (28), since dB3 /dc = dB2 /dc from Eq. (25). The derivative dB2 /dc is derived from Eq. (24), dD1 /dc from Eq. (19), and dyb /dc from Eq. (26). Appendix B.

(A1)

We know dy = ydt + ydz, from Eq. (1), which also gives: (dy)2 =  2 y2 dt. Substituting into Eq. (A1), we get: dV = Vy (ydt + ydz) + 0.5Vyy ( 2 y2 dt),

(A2)

Taking expectations, and given that E(dz) = 0, we get: E(dV ) = Vy ydt + 0.5Vyy ( 2 y2 dt),

(A4)

or E(dV ) = V  (y)ydt

+ 0.5V  (y) 2 y2 dt This is the return from the project (similar to capital gain). In addition, there is the continuous cash flow from the project (similar to dividend), given by (y)dt over this period. Thus, the total return from the project is given by: [E(dV) + (y)dt]/V. From the local expectations hypothesis, this should be equal to the discount rate

Here we present the model’s results with other sets of parameter values, taken from some well-known contingent-claim models in the capital structure literature. We start with the model of Hackbarth, Miao, and Morellec (2006), which should be representative of the corporate sector since they “. . .select parameter values that roughly reflect a typical S&P 500 firm” (page 531). Using their parameter values, we get the following inputs: r = 5.5%,  = 0.5%,  = 25%,  = 15%, and ˛ = 40% (and keeping v and Q unchanged at v = 1 and Q = 1). With these parameter values, the optimal leverage ratio with  = 0.05 is 38.43%, and the optimal leverage with  = 0.95 is 42.54%. Thus, the spread in leverage ratio over the range of price sensitivity is a little over 4%, not very far from our base-case scenario. Next, using the base-case parameter values of Sundaresan and Wang (2007), we have: r = 6%,  = 0,  = 15%,  = 25%, and ˛ = 35% (along with v = 1 and Q = 1). The results are: optimal leverage ratio

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is 59.69% for  = 0.05, and 65.55% for  = 0.95. The spread in optimal leverage ratio is almost 6%. Titman and Tsyplakov (2007) focus on the gold mining industry in their numerical simulations. They use parameter values that are “. . .chosen to roughly match empirical observations for selected firms in the gold mining industry” (page 418). Using their paper, we get the following parameter values: r = 3%,  = 1%,  = 10%,  = 35%, and ˛ = 50% (along with v = 1 and Q = 1). With these parameter values, the optimal leverage ratio comes to 68.06% for  = 0.05, and 71.89% for  = 0.95. The spread in optimal leverage ratio is about 3.8%. Finally, from Hackbarth and Mauer (2012), we get the following parameter values: r = 6%,  = 1%,  = 25%,  = 15%, and ˛ = 25% (along with v = 1 and Q = 1). The results are: optimal leverage ratio is 51.41% for  = 0.05 and 52.88% for  = 0.95, giving a spread in optimal leverage ratio of only 1.5%. It is clear that the effect of market power or price sensitivity () on optimal capital structure is a function of the values of the input parameters, particularly the tax rate and bankruptcy cost (not surprising, in the context of a “trade-off theory” model). Therefore, the importance of market power in capital structure decisions will likely vary from company to company, and it is not possible to make a blanket statement about its importance. However, after examining the results with parameter values from a number of well-respected models in the literature, we find that in most cases market power does seem to be a potentially important factor in leverage decisions. Thus, while the effect of market power on capital structure might be insignificant in some cases, in most cases it can make a substantial difference to the optimal capital structure. Hence, we argue that it is generally a good idea to include this factor in capital structure models. References Aguerrevere, F. L. (2003). Equilibrium investment strategies and output price behavior: A real-options approach. Review of Financial Studies, 16, 1239–1272. Bar-Ilan, A., & Strange, W. C. (1999). The timing and intensity of investment. Journal of Macroeconomics, 21, 57–77. Brander, J. A. T., & Lewis, R. (1986). Oligopoly and financial structure: The limited liability effect. American Economic Review, 76, 956–970. Brennan, M., & Schwartz, E. (1978). Corporate income taxes, valuation, and the problem of optimal capital structure. Journal of Business, 51, 103–114. Dixit, A. (1989). Entry and exit decisions under uncertainty. Journal of Political Economy, 97, 620–638. Dixit, A., & Pindyck, R. (1994). Investment under uncertainty. Princeton, NJ: Princeton University Press. Dobbs, I. M. (2004). Intertemporal price cap regulation under uncertainty. The Economic Journal, 114, 421–440. Faulkender, M., Flannery, M., Hankins, K., & Smith, J. (2007). Are Adjustment Costs Impeding Realization of Target Capital Structures? SSRN Working Paper.

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